The performance of LLP is summarized in Table REF . LLP achieves \(\mathcal {R}_T\!=\mathcal {O}(T^{\frac{3-\beta }{4}})\)  and \(\mathcal {V}_T\!=\mathcal {O}(T^{\frac{1+\beta }{2}})\)  for worst-case (or, no) predictions, which are tunable through parameter \(\beta \in [1/2,1)\) . For instance, with \(\beta \!=\!1/2\) , we obtain \(\mathcal {R}_T\mathcal {O}(T^{\frac{5}{8}})\)  and, \(\mathcal {V}_T=\mathcal {O}(T^{\frac{3}{4}})\) , that are further reduced to \(\mathcal {R}_T, \mathcal {V}_T\!=\mathcal {O}(\sqrt{T})\)  when \(-\mathcal {R}_T\!=\mathcal {O}(T^{\frac{1}{2}})\) , i.e., when \(\lbrace x_t\rbrace \)  does not outperform \(x^\star \)  by more than that. With perfect predictions, LLP achieves \(\mathcal {R}_T\!=\mathcal {O}(1)\) , \(\mathcal {V}_T\!=\mathcal {O}(T^{\frac{1+\beta }{2}})\)  which are tunable via \(\beta \in [0,1)\) ; while for linearly-perturbed constraints (as in [1]}) LLP ensures \(\mathcal {R}_T=\mathcal {O}(\sqrt{T})\)  and \(\mathcal {V}_T=\mathcal {O}(T^{5/8})\) . These results improve previously-known bounds for the general problem, i.e., without imposing additional assumptions such as strong convexity of functions and domains, or fixed \(T\) . Finally, they include as special cases the benchmarks with static or stationary constraints of [2]}, [3]}, [4]}, [5]}, [6]}. Importantly, unlike all prior constrained-OCO algorithms, the constant factors of \(\mathcal {R}_T\)  and \(\mathcal {V}_T\)  shrink proportionally to the predictions' accuracy, an advantage that is revealed even with simple numerical examples.
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